Certainly, let's break down the problem step-by-step to find the latent heat of fusion of the substance.
### Given Data:
- The amount of heat required to melt the substance ([tex]\( Q \)[/tex]): [tex]\( 3.44 \times 10^6 \)[/tex] joules (J)
- The mass of the substance ([tex]\( m \)[/tex]): [tex]\( 42.1 \)[/tex] kilograms (kg)
### Latent Heat of Fusion:
The latent heat of fusion ([tex]\( L_f \)[/tex]) is defined as the amount of heat required to change a unit mass of a substance from solid to liquid at its melting point without changing its temperature.
The formula to calculate the latent heat of fusion is:
[tex]\[ L_f = \frac{Q}{m} \][/tex]
Where:
- [tex]\( Q \)[/tex] is the heat absorbed or released (in joules, J)
- [tex]\( m \)[/tex] is the mass of the substance (in kilograms, kg)
### Step-by-Step Calculation:
1. Heat Given ([tex]\( Q \)[/tex]): [tex]\( 3.44 \times 10^6 \)[/tex] J
2. Mass of the Substance ([tex]\( m \)[/tex]): [tex]\( 42.1 \)[/tex] kg
3. Calculate latent heat of fusion [tex]\( L_f \)[/tex]:
[tex]\[ L_f = \frac{3.44 \times 10^6 \text{ J}}{42.1 \text{ kg}} \][/tex]
4. Perform the division:
[tex]\[ L_f = \frac{3440000 \text{ J}}{42.1 \text{ kg}} \][/tex]
[tex]\[ L_f \approx 81710.21377672208 \text{ J/kg} \][/tex]
### Conversion to Scientific Notation:
The calculated latent heat of fusion is approximately [tex]\( 81710.21377672208 \)[/tex] J/kg. We express this value in scientific notation:
[tex]\[ L_f \approx 8.17 \times 10^4 \text{ J/kg} \][/tex]
### Conclusion:
The latent heat of fusion of the substance is closest to the option:
[tex]\[ 8.17 \times 10^4 \text{ J/kg} \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{8.17 \times 10^4 \text{ J/kg}} \][/tex]
